Here, \(f\)  is the outer-level objective function, \(g\)  is the inner level objective function, and \(\mathcal {X} \in \mathbb {R}^{d_1}\)  denotes the outer-level input space. (REF ) is called bilevel optimization because the outer objective function \(f\)  depends on the solution to the inner objective \(g\) . Recently, BO has found a myriad of applications in modern machine learning: hyperparameter optimization [1]}, reinforcement learning [2]}, personalized federated learning [3]}, multi-task learning (MTL), representation learning [4]}, etc. We refer the reader to [5]} for a comprehensive survey of applications of BO in machine learning and computer vision.
